Centre of an algebra

Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra ({\em full centre}) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories. As an example we treat the case of group-theoretical categories.

💡 Research Summary

The paper introduces a construction called the “full centre” that assigns to any algebra A in a monoidal category 𝒞 a commutative algebra C(A) living in the monoidal centre Z(𝒞). The motivation comes from rational conformal field theory (RCFT), where one encounters algebras of bulk fields that are required to be commutative in a braided sense. The authors first define the full centre at the level of algebras: starting from the A‑A‑bimodule structure on A⊗A, they build a natural half‑braiding that makes C(A) an object of Z(𝒞) equipped with a multiplication that is commutative with respect to the braiding. This construction is canonical and functorial with respect to algebra homomorphisms.

A central result is the Morita invariance of the full centre. Two algebras A and B are Morita equivalent if their categories of left modules 𝓜_A and 𝓜_B are equivalent as 𝒞‑module categories. Using a suitable A‑B‑bimodule P (and its dual Q) the authors construct 𝒞‑module functors between 𝓜_A and 𝓜_B and show that these functors transport the full centre: C(A) and C(B) become isomorphic objects of Z(𝒞). The proof relies on the observation that the full centre can be described as the internal endomorphism algebra of the unit object in the module category, and that internal ends are preserved under equivalences of module categories.

The paper then lifts the construction from algebras to arbitrary 𝒞‑module categories 𝓜. For any such 𝓜 the authors define a full centre C_𝓜 as the internal endomorphism algebra of the unit of 𝓜, regarded as an object of Z(𝒞). When 𝓜 = 𝓜_A this recovers the previous definition, but the new viewpoint makes it clear that the full centre is a 2‑categorical invariant: if 𝓜 and 𝓜′ are equivalent as 𝒞‑module categories, then C_𝓜 ≅ C_𝓜′ in Z(𝒞). This result extends Morita invariance to the level of module categories and shows that the full centre is a robust invariant under higher‑categorical equivalences.

To illustrate the theory, the authors work out the full centre in the group‑theoretical setting. For the representation category Rep(G) of a finite group G, algebras correspond to G‑graded algebras or twisted group algebras k^α